Nonlinear Multivariate Function-on-function Regression with Variable Selection

This paper proposes a multivariate nonlinear function-on-function regression model, which allows both the response and the covariates can be multi-dimensional functions. The model is built upon the multivariate functional reproducing kernel Hilbert space (RKHS) theory. It predicts the response function by linearly combining each covariate function in their respective functional RKHS, and extends the representation theorem to accommodate model estimation. Further variable selection is proposed by adding the lasso penalty to the coefficients of the kernel functions. A block coordinate descent algorithm is proposed for model estimation, and several theoretical properties are discussed. Finally, we evaluate the efficacy of our proposed model using simulation data and a real-case dataset in meteorology.